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Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 43

Circuit Analysis- Basic

Concepts -



Circuit Analysis

DEVELOP TOOLS FOR THE ANALYSIS AND DESIGN OF BASIC ELECTRIC CIRCUITS



BASIC ANALYSIS STRATEGY

Note: Some Devices are NONlinear



Potential↔Flow Circuit

A “Circuit” describes the Relationship between• Indestructible-Entity

(Mass, Heat, Charge) Movement or FLOW

• Energy-Producing POTENTIAL (Gravity, Pressure, Voltage,Temperature)



Electro-Fluidic Analogy

Analogous Fluidic and Electrical Circuits

Examine Individual Circuit ElementsFluidic Electrical



Potential Source

Voltage is the Electrical PRESSURE

Pressure Source → Centrifugal Pump (with fluid slip)

Electrical Potential Source → Voltage Source



Flow (Mass Movement) Source

Current is the Electrical MASS-FLOW

Incompressible-Fluid Flow-Source → Positive Displacement Pump (NO fluid slip)

Electrical Current Source → Current Source



Conductors (move mass)

Perfect Conductors transport “Flow” with NO LOSS of Potential



Resistor (hinders flow)

Resistors CAUSE a Loss of Potential with Flow thru Them

Flow Resistor → Orifice-Restriction, or Changes in Direction

Electrical Resistor → Carbon-Cylinder or

wire-coil



Flow On/Off Control

Valves & Switches Turn Flow On/Off

Flow On/Off → ShutOff Valve; typically “Ball”, “Plug” or “Gate” type

Electrical Current On/Off → 2-Pole,

Single-Throw (2PST) Switch



Capacitor Accumulates Flow

Capacitors Produce a Flow-Change when Pressure Changes

dt

dPCm

PCm

f

f

dt

dvCi

VCQ

A Balloon would be a form of Fluidic Capacitance. The Units for the Fluidic Capacitance, Cf, are inverse m∙s2

The Units of Electrical Capacitance are A∙s/V



Inductor Resists FLOW Chg

Inductors Produce a Pressure-Change when Flow Changes

Velocity Fluid is

222

V

VVP NarwWide

mm

loPhiP

dt

mdLPPP flohi

hiv lov

i i

dt

diLvvv Llohi

Based on Bernoulli-Flow of an incompressible Inviscid Fluid. The Units for the Fluidic Inductance, Lf, are inverse meters (m−1)

The Units of Electrical Inductance are V∙s/A



Another Circuit Analogy

Resistors in “Series

• Note that “Flow” moves from HI-potential to LO-potential in BOTH Cases



Electric Circuit – What is it?

ELECTRIC CIRCUIT IS AN INTERCONNECTION OF ELECTRICAL COMPONENTS

+-

L

C

1R

2R

Sv

Ov

TY

PIC

AL L

INEA

R

CIR

CU

IT

a b2 TERMINAL COMPONENT

characterized by thecurrent through it andthe voltage differencebetween terminals

NODE

NODE



Basic Concepts

Learning Goals• System Of Units: The SI Standard System;

Prefixes• Basic Electrical Quantities: Charge,

Current, Voltage, Power, And Energy• Circuit Elements: Active and Passive

– Passive elements typically have TWO connections: INput & OUTput, and can only DISSIPATE POWER

– Active elements have at LEAST 3 conntions: Input, OUTput, and CONTROL



International System of Units (SI)

SI base units • The SI system of Units Is Founded On

Units For Seven Base Quantities Assumed To Be Mutually Independent, As Given Below SI Base Units

Base quantity Name Symbol

length meter m

masskilogram

kg

time second s

electric current ampere A

thermodynamic temperature

kelvin K

amount of substance mole mol

luminous intensity candela cd

http://physics.nist.gov/cuu/index.html

Only One “Electrical” Unit



SI DERIVED ELECTRICAL UNITS Other quantities, called derived quantities, are defined in terms of the seven

base quantities via a system of quantity equations. The SI units for these derived quantities are obtained from these equations and the 7 SI base units

Derived quantity Name Symbol

Expression in terms of

other SI units

Expressionin terms of

SI base units

Frequency hertz Hz - s-1

energy, work, quantity of heat joule J N·m m2·kg·s-2

power, radiant flux watt W J/s m2·kg·s-3

electric charge, quantity of electricity coulomb C - s·A

electric potential difference,electromotive force

volt V W/A m2·kg·s-3·A-1

capacitance farad F C/V m-2·kg-1·s4·A2

electric resistance ohm Ω V/A m2·kg·s-3·A-2

electric conductance siemens S A/V m-2·kg-1·s3·A2

inductance henry H Wb/A m2·kg·s-2·A-2



SI prefixes Factor Name Symbol Factor Name Symbol

1024 yotta Y 10-1 Deci d

1021 zetta Z 10-2 Centi c

1018 exa E 10-3 milli m

1015 peta P 10-6 micro µ

1012 tera T 10-9 nano n

109 giga G 10-12 pico p

106 mega M 10-15 femto f

103 kilo k 10-18 atto a

102 hecto h 10-21 zepto z

101 deka da 10-24 yocto y



Derived Units

One Ampere Of Current Carries One Coulomb Of Charge Every Second

sCA

CJV

• 1 Coulomb = 6.242x1018 e-

– e- The Charge On An Electron

A Volt Is A Measure Of EnergyPer Charge (Electrical Potential) • Two Points Have A Potential Difference Of

One Volt If One Coulomb Of Charge Gains One Joule Of Energy When It Is Moved From One Point To The Other.



Derived Units cont. The Ohm Is A Measure Of The

Resistance To The Flow Of Charge• There Is One Ohm Of Resistance If It

Requires One Volt Of Electrical-Pressure To Drive Through One Ampere Of Current

AV

AVW

One Watt Of Power Is Required To Drive One Ampere Of Current Against An Electromotive Potential Difference Of One Volt



Current And Voltage Ranges



Current = Charge-Flow Strictly Speaking Current (i) is a Basic

Quantity And Charge (q) Is Derived • However, Physically The Electric Current is

Created By Charged-Particle Movement

An Electric Circuit Acts a Pipeline That Moves Charge from One Pt to Another• The Time-Rate-of-Change for Charge

Constitutes an Electric Current; Mathematically

qtqditqdt

tdqti

t

OR



Water-Flow Analogy Water Particles in a Pipeline Move

• From HIGH-Pressure to LOW-Pressure In Current-Flow Electrical Charges

move• From HIGH-Potential (Voltage) to

LOW-Potential Direction Convention

• Almost all Solid-State Current Flow is Transported by ELECTRONS, Which Move from LOW Potential to HIGH



Fluid↔Current Flow Analogs

Think of the• Voltage as the

Electrical “Pressure”

• Current as the Electrical Fluid

• Wire as the Electrical “Pipe”

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imgele/ohmpoi.gif



Current & Electron Flow

These Alternatives Constitute Conventional POSITIVE Current Flow

+ +

+ +

q(t)

- -

- -

q(t)

Vhi Vhi

Vlo Vlo

• POSITIVE Charges Moving: Hi-v → Lo-v• NEGATIVE Charges Moving: Lo-v → Hi-v



Current/Charge Calculation

If The Charge is Given, Determine The Current By Differentiation

If The Current Is Known, Determine The Charge By Integration

EXAMPLE

+ +

+ +

q(t)

Vhi

Vlo]C[)120sin(104)( 3 ttq

A)120cos(120104 3 tdt

dqti

]mA[)120cos(480)(or tti



AnotherExample

0

00)( 2 tmAe

tti t

Find The Charge That Passes During In The Interval 0<t<1s

)2

1(

2

1

2

1 021

0

21

0

2 eeedeq

Find The Charge As A Function Of Time

t t

dedxxitq 2)()(

0)(0 tqt

t

tt

xx eedxetqt0

2

0

22 )1(2

1

2

1)(0

And the units for the charge?...

)1(2

1 2 eq Units?

>> t=[0:0.001:1]; % in Sec>> i_of_t = exp(-2*t); % in mA>> q = trapz(t, i_of_t) % in mCoul

q = 0.4323

>> syms x t>> q_of_t = int(exp(-2*x), 0, t) q_of_t = -1/2*exp(-2*t)+1/2

By MATLAB



GraphicalExample

1 2 3 4 5 610

102030

Charge(pC)

Time(ms)Here we are given thecharge flow as a functionof time.

)/(10100102

10101010 93

1212

sCs

Cm

DETERMINE THE CURRENT

1 2 3 4 5 610

102030

Time(ms)

) Current(nA

40

20

To determine current we must take derivatives.PAY ATTENTION TO UNITS

Zero currents



Convention For Currents It Is Absolutely

Necessary To Indicate The Direction Of Movement Of Charged ParticlesA Positive Value For

The Current Indicates Flow In The Direction Of The Arrow (The Reference Direction)

A Negative Value For The Current Indicates Flow In The OPPOSITE Direction Than The Reference Direction

The Universally Accepted Convention In Electrical Engineering is That Current is the Flow Of POSITIVE Charges• We Indicate Our

ASSUMED Direction Of Flow For Positive Charges– THE REFERENCE

DIRECTION



Double Index Notation

If The Initial And Terminal Node Are Labeled One Can Indicate Them As SubIndices For The Current Name

a bA5

AIab 5

a b

a

a

ab

b

b

A3

A3 A3

A3AIab 3

AIba 3

AIab 3

AIba 3

POSITIVE CHARGESFLOW LEFT-RIGHT

POSITIVE CHARGESFLOW RIGHT-LEFT

baab II



Example

This Example Illustrates The Various Ways In Which The Current Notation Can Be Used

b

a

I

A3

ab

cb

I

AI

AI

4

2

A2

c

3A

• To Find Iab Assuming that Charge is CONSERVED– More on this

later



Voltage Conventions

One Definition For the Volt unit • Two Points Have A Voltage Differential Of

One Volt If One Coulomb Of Charge Gains (Or Loses) One Joule Of Energy When It Moves From One Point To The Other • If The Charge GAINS

Energy Moving From a To b, Then b Has HIGHER Voltage Than a

• If It Loses Energy Then b Has LOWER Voltage Than a

a

b

C1



Voltage Conventions cont.

Dimensionally, the Volt is a Derived Unit

sA

mN

COULOMB

JOULEVOLT

Voltage Is Always Measured In a RELATIVE Form As The Potential DIFFERENCE Between Two Points• It Is Essential That Our Notation Allows Us

To Determine Which Point Has The Higher Electrical Potential (Voltage)



THE + AND - SIGNS DEFINE THE REFERENCEPOLARITY

V

POINT A HAS 2V MORE THAN POINT B

POINT A HAS 5V LESSTHAN POINT B

If The Number V Is Positive, Then Point A Has V Volts More Than Point B, If The Number V Is Negative, Then Point A Has |V| Less Than Point B



2-Index Notation for Voltages

Instead Of Showing The Reference Polarity We Agree That The FIRST SubIndex Denotes The Point With POSITIVE Reference Polarity

VVAB 2 VVAB 5 VVBA 5BAAB VV



Energy = Potential/Charge Voltage Is A Measure Of ENERGY Per Unit CHARGE

• Charges Moving Between Points With Different Voltage ABSORB or RELEASE Energy – They May Transfer Energy From One Point To Another

Converts chemical energy stored in the battery to thermal energy in the lamp filament which turns incandescent and glows

EQUIVALENT CIRCUIT

Charges gainenergy here

Charges supplyEnergy here

Battery supplies energy to charges. Lamp absorbs energy from charges.

The net effect is an energy TRANSFER



Example What Energy Is Required To Move 120

[C] From Point-B To Point-A?• Recall Potential = Work/Charge

JVQWQ

WV 240

VVAB 2

120 [C]

• Since The Charges Move To a Point With Higher Voltage –The Charges Gained (Or Absorbed) Energy



Example Recall Again Potential = Work/Charge

• Work = 5 J• Charge = 1C

The Voltage Difference Is 5V• Which Point Has

Higher Voltage?– VAB = 5V

V51[C]

]J[5

Q

WV

V5



Energy and Power Each Coulomb Of

Charge Loses 3[J] Or Supplies 3[J] Of Energy To The Element

The Element Receives Energy At A Rate Of 6[J/s]

The Electric Power Received By The Element Is 6[W]• How Do We Recognize If

An Element SUPPLIES Or RECEIVES Power?

2[C/s] PASSTHROUGH THE ELEMENT

VIP

dpttwt

t

2

1

)(),( 12

In General



Passive Sign Convention

Power Received (Absorbed/Dissipated) is Positive While Power SUPPLIED (Generated) is Considered NEGATIVE

a b

abV

abIababIVP

– If Voltage And Current Are Both Positive The Charges Move From High To Low Voltage, then The Component Receives, or Dissipates Energy – It Is A Passive Element



Passive Element Assumption

We Assume PASSIVE Elements• A Consequence Of This Convention Is That

The Reference Directions For Current And Voltage Are Not Independent– Given This Reference Potential-Polarity

a b

abV

Then the Reference Direction For Current



Passive Element Convention

The Voltage Polarity Given That The Reference Current Flows From Pt-A to Pt-B for the assumed Passive Element

a babI

+ −

• VA POSITIVE Relative to VB VAB > 0– For the PASSIVE Element



Example

The Element Receives 20W of Power• What is the Current?

Select ReferenceDirection Based OnPassive Sign Convention

]A[2or

)V10(]W[20

ab

ababab

I

IIV

a b

abV

abI

Vab = -10V

Given

• 2A of Current Flows RIGHT-to-LEFT ←




When In Doubt Label The Terminals Of The Component

1

2

1

2

W48or

A4V,12 1212

P

IV

Element SUPPLIES Power Element RECEIVES Power

W8or

A2V,4 1212

P

IV




Select Voltage Reference Polarity Based On CURRENT Reference Direction

VV

AVW

IVW

AB

AB

ABAB

4

)5(20

][20

][4VVAB

AI

IVW

IVW

AB

AB

ABAB

8

)5(40

][40

][8 AI

In These Examples the Unknown V & I have Polarity & Direction OPPOSITE to the Passive Assumption



Assigning The Reference If V Given Across the Element

• Iref Flows: Vhi→Vlo

IF I Given Thru the Element• Vref DROPS in The Current Direction

ABAB

ABABAB

VVVV

AVIVW

1and20

2][40

][201 VV ][5 AI

A

B

A

B

ABAB

ABABAB

IIAI

IVIVW

and5

10][50



Example Determine Power Absorbed By Each

Circuit Element; 1, 2, and 3• In this Passive Ckt, a single V-Source is an

Active, Power-SUPPLYING Element– Current Runs VLo→VHi; Sets I Direction

Assume that Voltage DROPS Across the Two Passive Elements is In the DIRECTION of Current Flow

+-

V24

V6

V18

A2

A2

123

WAVP 12)2)(6(1

WAVP 36)2)(18(2

WP

AVP

AVP

48

)2)(24(

)2)(24(

3

3

3

Note The Supplied-Received Power-Balance



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 43

-APPENDIX

-More ExamplesApplied Math



Done for Lecture/Discussion

Please Be Ready for a LAB during the Next Meeting.• Mr. Phillips will have something fun &

interesting to do

See you then… See Me to Add



Example Use Power Balance To Compute I0

AV

VAI

IVWW

IVWW

PP abs

166

6176182

6176323010812

0

0

0

sup

Thus a 1A Current Flows in the Direction Assumed in the Figure



Mathematical Analysis

Develop A Set Of Mathematical Equations That Represent The Circuit • A Mathematical Model

Learn How To Solve The Model To Determine Circuit Behavior (ENGR25)

This Course Teaches The Basic Techniques Used To Develop Mathematical Models For Electric Circuits



Linear Models The Models That Will Be Developed Have

“Nice” Mathematical Properties. • In Particular They Will Be Linear Which Means

That They Satisfy The Principle Of Superposition:

uTy

22112211 uTuTuuTy

– Model:

– Principle of SuperpositionGiven 1, 2, are CONSTANTS



Sophisticated Math Tools

This Course Applies the Tools Learned in MTH1 & ENGR25 to Solve Complex Physics/Math Circuit Models• For The First Part We Will Be Expected To

Solve Systems Of Algebraic Equations

20642

0164

84912

321

321

321

VVV

VVV

VVV

– Watch for These in MTH-6



Math Tools cont.• Later The Models Will Be Differential

Equations Of The Form

tfdt

tdfy

dt

dy

dt

yd

tfydt

dy

4384

3

2

2

– Technically These Are Linear, Nonhomogeneous, Ordinary, Constant Coefficient Differential Equations Of The 1st & 2nd Order Watch for them in MTH-4



Example A Camcorder Battery Plate Claims That

The Unit Stores 2700 mAHr At 7.2V • What Is The Total Charge And Energy Stored?

– Charge → The 2700 mAhr Specification Indicates That The Unit Can Deliver 2700 mA for ONE Hour

]C[1072.9orHr1Hr

s3600

s

C102700 33

QQ

][10998.6or][2.71072.9][ 43 JWJC

JVCQW

(0.02 kWhr)

– Stored Energy → The Charges Are Moved Thru a 7.2V Potential Difference

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